Translation functors and decomposition numbers for the periplectic Lie superalgebra $\mathfrak{p}(n)$

Abstract

We study the category ${\mathcal{F}}_n$ of finite-dimensional integrable representations of the periplectic Lie superalgebra $\mathfrak{p}(n)$. We define an action of the Temperley--Lieb algebra with infinitely many generators and defining parameter $0$ on the category ${\mathcal{F}}_n$ by translation functors. We also introduce combinatorial tools, called weight diagrams and arrow diagrams for $\mathfrak{p}(n)$ resembling those for $\mathfrak{gl}(m|n)$. Using the Temperley--Lieb algebra action and the combinatorics of weight and arrow diagrams, we then calculate the multiplicities of standard and costandard modules in indecomposable projective modules and classify the blocks of ${\mathcal{F}}_n$. We also prove that indecomposable projective modules in this category are multiplicity-free.

v5: two typos in Section 8.4 fixed, as well as a few other minor typos; affiliation fixed. v4: revised version (e.g. added suggestions of a referee, minor typos fixed, funding acknowledgement updated). v3: Title changed!!! Content is the same as in the previous version